Think about this. Suppose the question asked you to calculate the area inside the circle. How would the integral change? The two curves still intersect at the same points, so using your logic, you'd end up with same integral. Obviously, that can't be right. There's no reason to believe the area inside and outside the circle are the same.

I recommend you rethink the problem starting from the more general formula for the area
$$A = \iint r\,dr\,d\theta,$$ with the appropriate limits, and try to understand where the formula
$$A = \int \frac 12 r^2 \,d\theta$$ comes from. The latter is a special case of the first one, and you need to understand when you can actually use it.